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How many number of five digits can be formed without repetation if 2,3 and5 always occur in each number

How many number of five digits can be formed without repetation if 2,3 and5 always occur in each number

Grade:10

1 Answers

Arun
25750 Points
6 years ago
Let's think of 2 cases. 
1. A formed number 
has 2, 3 or 5 in 10 thousands' place-
Out of remaining 4 places, 2 places are fixed. Now, out of remaining 2 places, one can be filled with 7 digits ( except 2, 3 & 5) & other with 6 digits ( One which fills first unknown place is excluded.). 
Position of these 4 digits can be reversed in 4! ways. 
But doing this, same condition occurs twice. E.g. We have selected (6,7) & (7,6) as well.
So, we have to divide the total number by 2. 
As 5th place can be filled with 2, 3 or 5, we have
Numbers formed by condition 1 = 4!*7*6*3/2=63*4!
2. A formed number has no 2, 3 or 5 in its 5th place-
5th place can be filled by 6 digits  
(excluding 2,3,5 & 0). If 0 is in 5th place, it will be a 4 digited number. Remaining unknown place can be filled by 6 digits (excluding 2,3,5 & one which is in 5th place). 
If we keep 5th place fixed, other 4 digits can be reversed in 4! ways. 
Numbers formed by condition 2 = 4!*6*6
Total numbers = 4!*63+4!*36
                            =24×99=2376.

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